Methods for producing lens arrays

ABSTRACT

A lens array for imaging image elements in an object plane, and a method of making a lens array. The lens array includes lenslets formed in or on one side of a transparent or translucent material with the image elements disposed on the opposite side, and has a gauge thickness corresponding to the distance from the apex of each lenslet to the object plane. Each lenslet has a set of lens parameters. The gauge thickness and/or at least one lens parameter is or are optimised such that each lenslet has a focal point size in the object plane which is either substantially equal to the size of the image elements in the object plane, or varies from the size of the image elements by a predetermined amount.

PRIORITY CLAIM

This patent application is a continuation of U.S. patent applicationSer. No. 13/203,070; filed 26 Oct. 2011, which is a U.S. National Phaseof International Patent Application No. PCT/AU2010/000243, filed 3 Mar.2010, which claims priority to U.S. Provisional Patent Application No.61/157,309, filed 4 Mar. 2009, the disclosures of which are incorporatedherein by reference in their entirety.

FIELD

The disclosed embodiments relate to methods for designing andmanufacturing lens arrays, and to lens arrays produced thereby.

BACKGROUND

Lens arrays allow the generation of a number of different types ofoptical effect. For example, an array of lenses focussed on underlyingimage elements at the focal plane of the array can generate integralimages which appear to be three dimensional, which move, changemagnification or morph as the viewing angle changes, or which haveapparent depth outside the plane of the lens array. A further effect canbe achieved by interleaving two or more images underneath the lenses,for example in strips underneath a plurality of cylindrical lenses, sothat the viewer sees different images as the viewing angle changes. Suchvisual effects are useful in a number of applications includingdisplays, promotional materials, collectable articles and as opticallyvariable devices in security documents.

Lens arrays are generally manufactured from transparent polymericmaterials to produce a sheet of material, referred to herein as alenticular sheet. The pattern of the lenslets forming the array isembossed or otherwise formed on one side of the sheet, and the opposingside of the sheet is formed into a flat, generally gloss-type surface.The image elements are applied to or placed at the flat surface and maybe formed, for example, by printing or by a laser marking process. Thesheet material is commonly manufactured as a monolayer, but multilayermethods are also used.

The image elements may comprise printed dots. In one process, prior toprinting, a continuous image representing the desired final print on theflat surface is converted to a halftone image. After printing, thehalftone image will appear as a plurality of printed dots on the flatsurface.

The thickness (usually termed the gauge thickness) of the lenticularsheet has traditionally been determined by the focal length of thelenslets, such that incoming light rays substantially focus at the flatsurface of the sheet. This design is chosen to take advantage of theso-called sampling effect. The sampling effect ensures that a dotprinted at the focal length of the lens will appear to an observer at aparticular viewing angle as a line across a cylindrical lens, and willappear to fill the entire lens area for a non-cylindrical lens.Therefore, an observer cannot distinguish two adjacent dots inside asingle lens at a particular viewing angle.

In some cases, the material thickness and lens frequency (or pitch) ofthe lenslets may be pre-selected according to the needs of the endproduct, as well as the gauge limitations of the sheet materialmanufacturing process. The lens radius of curvature is then determinedbased on additional parameters such as the refractive index and Abbenumber of the polymeric material used, so as to focus incoming lightrays substantially at the flat side of the sheet.

The recent trend in the art has been to produce thinner lenticularsheets so as to reduce manufacturing costs, while at the same timebroadening the potential applications of optical effect articles.However, a thinner lenticular sheet generally requires a higher lensfrequency in order to produce a focussed image. For example, a materialproduced with a gauge thickness of 85 microns in polyester would requirea lens frequency of approximately 224 lenslets per centimetre. Printingoptical effect imagery on these high frequency microlens arrays isparticularly challenging, and severely limits the type of effect thatcan be achieved and the type of press and prepress methods than can beused. Furthermore, a high rate of waste material often results as veryhigh line screens must be utilised and very accurate colour-to-colourregistration becomes critical. These problems have meant that use ofvery high frequency lenticular sheet material has heretofore beenlimited.

One attempt to overcome the above problems is described in U.S. Pat. No.6,833,960. Lenses are formed as hemispheres on a substrate using curableresins in a printing press. In this method, it is not possible to formthe lenses on the substrate at their focal point. The lenses are thussubstantially off-focus and this nullifies the sampling effect. Theimagery produced by the method is thus substantially blurred.

Another method is described in U.S. Pat. No. 6,989,931 and includes acomposite image comprised of printed stripes viewable through alenticular screen from a first angle, with an object or image placed adistance behind the lenticular screen viewable through transparentstripes at a second angle. In one embodiment, a lenticular materialthinner than its focal length can be used since the stripes printed onthe flat side of the lenticular material do not contain multiplexedoptical effect imagery. The lenslets need not possess the resolvingpower required for traditional optical effect imagery, but instead viewonly one half of the area behind each lenslet. However, use of thismethod for multiplexed lenticular imagery or intricate moiré effects canproduce severe blurring.

Accordingly, there is a need for a method of reducing the gaugethickness of a lens array without introducing substantial blurring orother objectionable image artifacts.

In some circumstances it may be desirable to manufacture a lenticularsheet at a particular gauge thickness. If this is the case, it may thenbe desirable to reduce the lens frequency, i.e. to increase the width ofeach lenslet, in order to maintain image quality in view of theconstraints of the printing process (or other process for forming theimage elements) to be used. It is therefore desirable to provide a lensarray and a method which allows a lower lens frequency to be usedwithout introducing substantial blurring or other objectionable imageartifacts.

Any discussion of documents, acts, materials, devices, articles or thelike which has been included in the present specification is solely forproviding a context for the disclosed embodiments. It is not to be takenas an admission that any or all of these matters form part of the priorart base or were common general knowledge in the field relevant to thedisclosed embodiments as they existed in Australia before the prioritydate of each claim in this application.

SUMMARY

One disclosed embodiment provides a lens array for imaging a pluralityof image elements in an object plane, the lens array including aplurality of lenslets formed in or on one side of a transparent ortranslucent material with the image elements disposed on the oppositeside, the lens array having a gauge thickness corresponding to thedistance from the apex of each lenslet to the object plane, wherein eachlenslet has a set of lens parameters, with the gauge thickness and/or atleast one lens parameter optimised such that each lenslet has a focalpoint size in the object plane which is substantially equal to the sizeof the image elements in the object plane, or which varies from the sizeof the image elements by predetermined amount.

Another embodiment provides a method of manufacturing a lens array forimaging a plurality of image elements in an object plane, the lens arrayincluding a plurality of lenslets, the lens array having a gaugethickness corresponding to the distance from the apex of each lenslet tothe object plane, the method including the steps of:

determining a scale parameter which is representative of the size of theimage elements in at least part of the object plane,

using the scale parameter to optimize the gauge thickness and/or atleast one parameter of a set of lens parameters for each lenslet,

and forming the lens array with the gauge thickness and the lensparameters in or on one side of a transparent or translucent materialwith the image elements being disposed on the opposite side of thetransparent or translucent material,

whereby, the lenslets have a focal point size in the object plane whichis substantially equal to the size of the image elements, or whichvaries from the size of the image elements by a predetermined amount.

The set of lens parameters may include lens width, refractive index, sagheight, radius of curvature, conic parameter and Abbe number. Some orall of these may be varied in order to obtain a focal point size withthe desired characteristics in the object plane.

Optionally, each lenslet has a cross-section which is a conic section.The lenslets may be cylindrical or have a part-spherical or asphericalcross-section. Each lenslet may be rotationally symmetric in the planeof the lens array. In one embodiment, each lenslet may be an elongatelenticule having substantially uniform cross section along its length.

DEFINITIONS

Focal Point Size H

As used herein, the term focal point size refers to the dimensions,usually an effective diameter or width, of the geometrical distributionof points at which rays refracted through a lens intersect with anobject plane at a particular viewing angle. The focal point size may beinferred from theoretical calculations, ray tracing simulations, or fromactual measurements. The present inventors have found that ray tracingsimulations using software such as ZEMAX closely match directmeasurements of lenses designed according to the methods describedherein. The ray tracing simulation can be adjusted to account for thefact that incoming rays are not exactly parallel in reality.

Focal Length f

In the present specification, focal length, when used in reference to amicrolens in a lens array, means the distance from the vertex of themicrolens to the position of the focus given by locating the maximum ofthe power density distribution when collimated radiation is incidentfrom the lens side of the array (see T. Miyashita, “Standardization formicrolenses and microlens arrays” (2007) Japanese Journal of AppliedPhysics 46, p 5391).

Gauge Thickness t

The gauge thickness is the distance from the apex of a lenslet on oneside of the transparent or translucent material to the surface on theopposite side of the translucent material on which the image elementsare provided which substantially coincides with the object plane.

Lens Frequency and Pitch

The lens frequency of a lens array is the number of lenslets in a givendistance across the surface of the lens array. The pitch is the distancefrom the apex of one lenslet to the apex of the adjacent lenslet. In auniform lens array, the pitch has an inverse relationship to the lensfrequency.

Lens Width W

The width of a lenslet in a microlens array is the distance from oneedge of the lenslet to the opposite edge of the lenslet. In a lens arraywith hemispherical or semi-cylindrical lenslets, the width will be equalto the diameter of the lenslets.

Radius of Curvature R

The radius of curvature of a lenslet is the distance from a point on thesurface of the lens to a point at which the normal to the lens surfaceintersects a line extending perpendicularly through the apex of thelenslet (the lens axis).

Sag Height s

The sag height or surface sag s of a lenslet is the distance from theapex to a point on the axis intersected by the shortest line from theedge of a lenslet extending perpendicularly through the axis.

Refractive Index n

The refractive index of a medium n is the ratio of the speed of light invacuo to the speed of light in the medium. The refractive index n of alens determines the amount by which light rays reaching the lens surfacewill be refracted, according to Snell's law:

n ₁*Sin(α)=n*Sin(θ),

where α□ is the angle between □ an incident ray and the normal at thepoint of incidence □ at the lens surface □□α is the angle between therefracted ray and the normal at the point of incidence, and n₁ is therefractive index of air (as an approximation n₁ may be taken to be 1).

Conic Constant P

The conic constant P is a quantity describing conic sections, and isused in geometric optics to specify spherical (P=1), elliptical (0<P<1,or P>1), parabolic (P=0), and hyperbolic (P<0) lens. Some references usethe letter K to represent the conic constant. K is related to P viaK=P−1.

Lobe Angle

The lobe angle of a lens is the entire viewing angle formed by the lens.

Abbe Number

The Abbe number of a transparent or translucent material is a measure ofthe dispersion (variation of refractive index with wavelength) of thematerial. An appropriate choice of Abbe number for a lens can help tominimize chromatic aberration.

Security Document

As used herein, the term security document includes all types ofdocuments and tokens of value and identification documents including,but not limited to the following: items of currency such as banknotesand coins, credit cards, cheques, passports, identity cards, securitiesand share certificates, driver's licences, deeds of title, traveldocuments such as airline and train tickets, entrance cards and tickets,birth, death and marriage certificates, and academic transcripts.

Transparent Windows and Half Windows

As used herein the term window refers to a transparent or translucentarea in the security document compared to the substantially opaqueregion to which printing is applied. The window may be fully transparentso that it allows the transmission of light substantially unaffected, orit may be partly transparent or translucent partially allowing thetransmission of light but without allowing objects to be seen clearlythrough the window area.

A window area may be formed in a polymeric security document which hasat least one layer of transparent polymeric material and one or moreopacifying layers applied to at least one side of a transparentpolymeric substrate, by omitting least one opacifying layer in theregion forming the window area. If opacifying layers are applied to bothsides of a transparent substrate a fully transparent window may beformed by omitting the opacifying layers on both sides of thetransparent substrate in the window area.

A partly transparent or translucent area, hereinafter referred to as a“half-window”, may be formed in a polymeric security document which hasopacifying layers on both sides by omitting the opacifying layers on oneside only of the security document in the window area so that the“half-window” is not fully transparent, but allows some light to passthrough without allowing objects to be viewed clearly through thehalf-window.

Alternatively, it is possible for the substrates to be formed from ansubstantially opaque material, such as paper or fibrous material, withan insert of transparent plastics material inserted into a cut-out, orrecess in the paper or fibrous substrate to form a transparent window ora translucent half-window area.

Opacifying Layers

One or more opacifying layers may be applied to a transparent substrateto increase the opacity of the security document. An opacifying layer issuch that

LT<L0, where L0 is the amount of light incident on the document, and LTis the amount of light transmitted through the document. An opacifyinglayer may comprise any one or more of a variety of opacifying coatings.For example, the opacifying coatings may comprise a pigment, such astitanium dioxide, dispersed within a binder or carrier of heat-activatedcross-linkable polymeric material. Alternatively, a substrate oftransparent plastic material could be sandwiched between opacifyinglayers of paper or other partially or substantially opaque material towhich indicia may be subsequently printed or otherwise applied.

In one disclosed embodiment, the gauge thickness of the lens array maybe optimised with respect to the size of the image elements and the setof lens parameters.

In another disclosed embodiment, the lens parameters may be optimisedwith respect to the size of the image elements and the gauge thickness.

By choosing lens parameters such that the focal point size is correlatedwith the size of the image elements, the thickness of the lens array, orthe lens frequency, can be reduced without substantially sacrificingimage quality. This is because the majority of rays refracted throughthe lenslets and reaching the object plane will still intersect with theregion covered by an image element at the desired viewing angle orangles and this allows the sampling effect to be maintained.

The thickness of the lens array can be reduced to provide a thinnerlenticular sheet which still results in high quality image effects.Alternatively, the thickness can be maintained whilst widening thelenslets to allow more printing to be included under each lenslet, thusimproving image quality and/or allowing for more complex visual effectsto be produced.

Optionally, the thickness of the lens array is less than the focallength of all of the lenslets.

In one disclosed embodiment, the predetermined amount by which the focalpoint size varies from the size of the image elements is not more than20% of the size of the image elements. If the focal point size is largerthan the size of the image elements, this allows for an even thinnerlenticular sheet whilst substantially maintaining the desired imagequality, since in general, only a relatively small portion of the powerdensity distribution of the refracted rays will lie at the edges of thespot. If the focal point size is slightly smaller, the transitionsbetween the image components producing the image effects can be madesmoother.

The image elements may take the form of dots, lines or other shapes. Theimage elements may be applied to a surface in the object plane on theopposite side of the transparent or translucent material in a variety ofways, including laser marking. In one disclosed embodiment, the imageelements are printed on the surface in the object plane. The method ofthe disclosed embodiments may include the step of applying a pluralityof printed dots to a rear surface of the transparent or translucentmaterial with the lenslets formed on a front surface to form anoptically variable device or article. Alternatively, a plurality ofprinted dots may be applied to a substrate (for example, of fibrous orpolymeric material), and the substrate attached to the rear surface ofthe transparent or translucent material.

The lenslets may formed by an embossing process in a transparent ortranslucent radiation-curable material applied to a substrate. Thetransparent or translucent radiation-curable material may be curedfollowing embossing, but is optionally embossed and cured substantiallysimultaneously. The substrate may be formed from a transparent ortranslucent polymeric material with the combined thickness of thesubstrate and radiation curable material corresponding to the gaugethickness of the lens array. In one disclosed embodiment, the substrateis a flexible, sheet-like structure, and the substrate and radiationcurable material form part of a security document, such as a banknote,credit card or the like. The substrate may be substantially the samerefractive index as the lenslets.

In one disclosed embodiment, the set of lens parameters is the same foreach lenslet.

In another disclosed embodiment, the focal point size, when averagedover at least two directions within the lobe angle of the lenslet, issubstantially equal to the size of the image elements, or varies fromthe size of the image elements by a predetermined amount.

The directions over which the focal point size are averaged may includethe on-axis direction, and an off-axis direction near the edge of thelobe angle.

Another disclosed embodiment provides a method of designing a lens arrayfor imaging a plurality of image elements in an object plane, the lensarray including a plurality of lenslets and having a gauge thicknesscorresponding to the distance from the apex of each lenslet to theobject plane, the method including the steps of:

estimating a scale parameter which is representative of the size of theimage elements in the object plane,

selecting a set of lens parameters for each lenslet, and

designing the lens array using the scale parameter to optimize the gaugethickness and/or at least one lens parameter of the set of lensparameters for each lenslet, wherein each lenslet has a focal point sizein the object plane which is substantially equal to the size of theimage elements, or which varies from the size of the image elements by apredetermined amount.

Optionally, the thickness of a lens array including the lenslets is lessthan the focal length of all of the lenslets.

The set of lens parameters may be the same for each lenslet.Alternatively, the lenslets in an area or areas of the lens array mayhave different lens parameters to the lenslets in the remainder of thelens array.

Optionally, the method further includes the step of measuring the sizesof the image elements in at least part of the object plane, wherein thescale parameter is estimated from the measured sizes of the imageelements. The measurement may be performed using a densitometer, oralternatively may be performed by directly measuring the sizes of theimage elements. Optionally, the image elements are part of a calibrationtemplate. In one disclosed embodiment, the image elements are printedlines or dots.

Measuring the sizes of the printed lines or dots allows the lens designto be tailored to the actual characteristics of the print, which maydepend on the type of printing apparatus, inks and other materials, andprepress equipment used.

The scale parameter may be estimated by computing the mean or maximum ofthe sizes of the image elements. Alternatively, it may be estimatedusing a robust estimator, optionally an M-estimator, or one of themedian, upper quartile or interquartile mean of the sizes of the imageelements.

A further disclosed embodiment provides a method of manufacturing anoptically variable device, including the steps of:

providing a substrate;

applying image elements to the substrate, the image elements beinglocated in an object plane;

determining a scale parameter which is representative of the size of theimage elements; and

forming a plurality of lenslets in a transparent or translucent materialon the substrate;

wherein each lenslet has a set of lens parameters determined so that thelenslets have a focal point size in the object plane which issubstantially equal to the size of the image elements, or which variesfrom the size of the image elements by a predetermined amount.

In one disclosed embodiment, the scale parameter is determined bymeasuring the sizes of the image elements.

Optionally, the substrate is formed from a transparent or translucentsheet-like material with the lenslets formed in or on a first surface onone side of the substrate and the image elements applied to a secondsurface on the opposite side of the substrate. The lenslets may beformed in the transparent or translucent sheet-like material itselfAlternatively, the lenslets may be formed in a transparent ortranslucent layer, e.g. by embossing a radiation-curable transparent ortranslucent resin, applied to a substrate which may be transparent,translucent or opaque.

The image elements may be formed by any convenient process, includingprinting or laser marking. In one disclosed method, the image elementsare printed dots.

Another disclosed embodiment provides an optically variable device,including a substrate and a plurality of lenslets formed in or on thesubstrate, and a plurality of image elements located in an object planein or on the substrate, wherein each lenslet has a set of lensparameters determined so that the lenslets have a focal point size inthe object plane which is substantially equal to the size of the imageelements, or which varies from the size of the image elements by apredetermined amount.

Optionally, the lenslets are part of a lens array having a gaugethickness which is less than the focal length of each of the lenslets.

A further disclosed embodiment provides an optically variable deviceincluding a lens array according to one disclosed embodiment.

An optically variable device manufactured by the methods above may beapplied to a wide range of articles, though the disclosed embodimentshave particular application to the field of security documents, and moreparticularly to security documents and articles formed from a flexiblesheet-like substrate, such as banknotes or the like. The opticallyvariable device may be formed in a window or half-window area of thesecurity document.

BRIEF DESCRIPTION OF THE DRAWINGS

Disclosed embodiments will now be described, by way of non-limitingexample only, with reference to the accompanying figures, in which:

FIG. 1 shows a cross section through a lens array of prior art design;

FIG. 2 shows one embodiment of the lens array;

FIG. 3 shows a second embodiment of the lens array;

FIGS. 4 to 6 depict the on- and off-axis focal point sizes of threelenslets according to different embodiments;

FIG. 7 shows the input and output dot shapes of two different printingprocesses;

FIG. 8 shows the power density distributions of a prior art lenslet anda lenslet according to one embodiment, respectively;

FIG. 9 shows incident light rays being refracted through a lensletaccording to one embodiment;

FIGS. 10 (a) and 10(b) depict the geometry of a lenslet;

FIGS. 11( a) to 11(d) illustrate a schematic cross-sectional viewthrough an article incorporating a lens array and image elements and theintermediate steps by which the article is formed;

FIGS. 12( a) to 12(c) show a schematic cross-sectional view through anarticle similar to that of FIG. 11( d), made by a modified method;

FIGS. 13( a) to 13(d) illustrate a schematic cross-sectional viewthrough an alternative article incorporating a lens array and imageelements, and the intermediate steps by which the article is formed;

FIGS. 14 and 15 are block diagrams showing two different embodiments ofa lens array manufacturing process for producing the articles of FIGS.11 to 13;

FIG. 16 is a set of interleaved printed image elements of an examplelenticular device; and

FIGS. 17( a) and 17 (b) illustrate the (simulated) relative illuminationof points along the width of the image elements of the device of FIG.16, when viewed (a) on-axis and (b) off-axis.

DETAILED DESCRIPTION OF DISCLOSED EMBODIMENTS

Referring initially to FIG. 1, there is shown a lens array 20 of priorart design, having a gauge thickness t, in which lenslets 22 having awidth W and substantially spherical profile focus incident rays 28 a and28 b onto the black dots 26 a and the white dots 26 b respectively. Thedots have been printed on lower surface 24. The thickness t issubstantially equal to the focal length of the lenslets, and so thefocal point size 30 is at a minimum.

The focal point size 30 of the prior art lenslet is smaller than theresolution of the printing at the lower surface 24. For example,traditional lenticular offset lithographic methods print an averagehalftone dot size of approximately 25 microns. A properly designedlenticular lens of 254 microns in width will collimate light to a focalpoint size of approximately five microns on axis, which is substantiallysmaller than the size of the printed dots 26 a, 26 b.

Referring now to FIG. 2, there is shown a lens array 120 according toone disclosed embodiment. Incident rays 128 a, 128 b are refractedtowards dots 26 a, 26 b respectively. The lens array 120 has a thicknesst′ which is less than t, and the lenslets 122 have a width W. Thelenslets 122 are designed in such a way that the focal point sizes 130a, 130 b are substantially equal in extent to dots 26 a, 26 b. We havefound that as long as the focal point size does not exceed the averagewidth of a printed halftone dot by more than 20%, the quality of theimage is not compromised. We have also found that simply producing anarbitrary non-focussing design severely degrades the image quality,resulting in an objectionably blurred image. The focal point size mayalso be slightly smaller than the average width, optionally no more than20% smaller.

FIG. 3 depicts an alternative lens array design, in which lens array 220is at the same thickness t as prior art lens array 20, but the width Wof the lenslets 222 has been increased. At the same time, other lensparameters have been varied so that incident rays 228 a, 228 b arerefracted and arrive at the object plane 224 to intersect with dots 26a, 26 b so that the focal point sizes 230 a, 230 b are againsubstantially equal in extent to the dots 26 a, 26 b. For example, thelens radius of curvature can be made larger, as shown in FIG. 3,possibly simultaneously varying other lens parameters such as refractiveindex, conic parameter or Abbe number in order to achieve the optimumimage quality.

FIG. 4 illustrates a cross sectional side view of a ray trace of a wideangle lenslet 105 designed according to one embodiment. Rays 102refracted at surface 101 reach the object plane 104 and result in focalpoint sizes 103A, 103B. In this embodiment, the focal point sizes ofon-axis points 103A and off-axis points 103B have been weighted equallyacross the entire viewing angle of the lenslet, this angle also beingknown as the lobe angle.

FIG. 5 shows a cross sectional side view of a ray trace of analternative wide angle lenslet 101 where the focal point width issubstantially equal to the average width of printed halftone dots 109A,across the lobe angle. In FIG. 6 there is shown a cross sectional sideview of a ray trace of a further wide angle lenslet where the printedhalftone dots 109A are larger still, allowing for a further reduction inmaterial thickness, or a coarser lens frequency, or both.

In FIG. 7 the top row of dots 110 represents digital pixels of a knownwidth 109B on a print calibration form that are output to a printingplate. Row 111 illustrates the printed result of printing row 110, wherenoticeable dot gain results in an average dot width 109C. Row 112illustrates the printed result of printing row 110 using another printmethod, where the dot gain is even greater than row 111, resulting inaverage dot width 109D. In this illustration, a different lens designmay apply for the printed dots in row 111 than the printed dots in row112, where the optimised lens design for row 111 may resemble FIG. 5,and the optimized lens design of row 112 may more closely resemble FIG.6.

Referring now to FIG. 8, there is shown a projection view of a printedhalftone dot 109C imaged by a lenslet according to one embodiment. FIG.8( a) shows contours 250 of the power density distribution 255 whichwould result if the object plane was located at the focal plane 252 ofthe array to produce spot 256 (FIG. 9). Instead, the spot 266 in objectplane 262 is larger than the dot 109C, but has a power densitydistribution 260, 265 such that the majority of the incident radiationis still reaching dot 109C in order to preserve the sampling effect.

In order to print a continuous tone image to paper or syntheticmaterials such as plastics, it is necessary to convert it to a halftoneimage. A number of methods for doing so are known in the art. Suchmethods represent continuous tone through the use of binary dots ofeither varying sizes, so called amplitude modulation (AM) methods, ordots of the same size with varying frequency, so called frequencymodulation (FM) methods. Various combinations of the two methods, calledhybrids, are also used. For the present purposes, any of these methodsmay be employed. However, the FM method, in its variety of formsincluding but not limited to dithering, error diffusion, or random orstochastic screening, is the preferred method because the dots remaingenerally constant in size.

Measurement of the characteristics of the printed halftone dots can beaccomplished using a variety of known methods. For example, the averagedot size can be determined by printing a press calibration templateconsisting of swatches of dots of a given size and having variousdensities, where each swatch typically represents a density value fromone percent to ninety nine percent. The template is subsequently imagedto film or plate, and printed onto the smooth side of an optical effectsubstrate. The printed result is then scanned using a densitometer, orsimilar tool, to determine the printed dot size.

Alternatively, the average dot size can be measured directly, forexample using a microscope fitted with a reticle displaying incrementsof measurement. In the direct method, a sample of dots can be measuredin each tonal value range, recorded, and their sizes averaged.

We have found that measurement of dots at a tonal value of about 20%provides the best results.

In some instances it may not be possible or feasible to obtain the abovemeasurements, because of varying press conditions or the like. If thisis the case then an average expected dot size can be estimated, fromprevious experience or otherwise.

With reference to FIG. 10, we now describe one method of optimising thedesign of a lenslet for use in one disclosed embodiment. For thepurposes of this non-limiting example, we take lenslet 300 to be anaspheric lens which is rotationally symmetric. This method relies onrelatively simple theoretical calculations using geometrical optics, andignoring edge effects at the periphery of the lens. The skilled personwill appreciate that many other methods are possible, including the useof more sophisticated physical models, ray tracing simulations and soon.

In FIG. 10( a), an image element in the form of a printed dot 305 offull width H and half-width h is located in an object plane an unknowndistance t (the gauge thickness) from the origin of the (x,y) coordinatesystem, which corresponds with the vertex 310 of the lens 300. Lens 300has a sag height s and a half-width w, and refractive index n (not shownin Figure). An optimal lens design will result in incident ray 320 whichis incident parallel to the x axis, arriving at the edge 315 of lens 300and refracted at an angle β intersecting the top of the dot 305. We thuswish to find an expression for t in terms of the lens parameters and thehalf-width h which is a scale parameter representative of the size ofthe dot 305.

The equation of the lens profile function y(x) is given by

P*x ²−2*R*x+y(x)²=0,

where R is the lens radius at the edge 305 of the lens, and P is theconic constant and is equal to 1−e², where e is the eccentricity. Inprinciple, a more general lens profile function y(x) including higherpowers of x could be chosen. However, it is generally more convenientfor lens design purposes to use the quadratic form of y(x) as above.

The normal 330 to the surface of the lens at edge 305 (x=s, y=w) has aslope

${m(x)} = \frac{- 1}{y^{\prime}(x)}$

where y′(x) is the first derivative of y(x). This slope is equal toTan(α) where □□ is the angle between incident ray 320 and the normal330, and so

Tan(α)=m(x)

so that

${\alpha (x)} = {{{ArcTan}\left( \frac{\sqrt{{2*R*x} - {P*x^{2}}}}{{P*x} - R} \right)}.}$

By Snell's law,

n ₁*Sin(α)=n*Sin(θ)

where θ is the angle between refracted ray 320′ and the normal 330, andn₁ is the refractive index of air (taken to be 1 as an approximationhereafter). Hence

$\theta = {{{ArcSin}\left( \frac{{Sin}(\alpha)}{n} \right)}.}$

The slope A of the line joining (s,w) and (t,h) is

A=−Tan(β)

and substituting for β=αθ,

$\begin{matrix}{A = {- {{{Tan}\left\lbrack {{\alpha (s)} - {{ArcSin}\left( \frac{{Sin}\left( {\alpha (s)} \right)}{n} \right)}} \right\rbrack}.}}} & (1)\end{matrix}$

It is relatively straightforward to show that t can be written as

$\begin{matrix}{t = {s + \frac{h - w}{A}}} & (2)\end{matrix}$

with A as in Eq (1) above and

$\begin{matrix}{{\alpha (s)} = {{{ArcTan}\left( \frac{w}{\sqrt{R^{2} - {P*w^{2}}}} \right)}.}} & (3)\end{matrix}$

The thickness t can be optimised with respect to one or more of the lensparameters R, n, P, w and s in the usual way, i.e. by taking the partialderivatives of the expression in Eq (2) with respect to one or more ofthose parameters and setting the partial derivatives equal to zero. Theresulting system of equations can be solved analytically or numericallyin order to find the set of lens parameters which gives the optimal lensthickness.

The optimisation may be a constrained optimisation. For example, theremay be practical manufacturing limitations on the range of t and it thusmay be desirable to limit t to that range of values. Constrainedoptimisation methods are known in the art.

The above formulae have been derived for incident rays parallel to the xaxis. The derivation can be generalised for off-axis rays 340, 350 andoff-axis dots (FIG. 10( b)) whereby one obtains

D=∥(M−m)*(t−s)+2*w∥,

where D is the off-axis dot size, M is the slope of the refracted ray340′ at one edge 345 of the lenslet, and m is the slope of the refractedray 350′ at the opposite edge 355 of the lenslet, with t being thedesired gauge thickness, s the sag height and w the lens half-width asbefore.

When the angle of deviation, δ, of the incident rays is zero, M=−m=A,and the equation reduces to

D=2M*(t−s)+2w.

In this case, D becomes equal to 2h, the full dot size, and

${t = {s + \frac{h - w}{A}}},$

which corresponds to the expression for on-axis rays derived earlier.

Alternatively to the above, it is possible to optimise the lenshalf-width w as a function of some or all of R, n, P and s, while t maybe held fixed. This can be done by rewriting Eq (2) in terms of w asfollows:

w=h−A*(t−s).

If t is held fixed, a constrained optimisation may be performed to findthe optimal lens half-width w.

As a further alternative, other lens parameters R, n, P or s may beoptimised in a similar manner to the above.

The above model does not explicitly include a treatment of chromaticaberration. The skilled person will appreciate that the conic constant Pand/or the Abbe number of the lens may be chosen to minimise chromaticaberration.

In FIG. 11( d) there is shown an article 400 formed from a substrate 410of transparent or translucent material having a lens array 420 formed ona front surface 411 on one side of the substrate 410 and image elements426 a, 426 b formed on a rear surface 412 on the opposite side of thesubstrate 410. In one disclosed method of manufacturing the article 400,the image elements 426 a, 426 b are first applied to the rear surface412 of the substrate 410 on the opposite side (FIG. 11( a)). The imageelements 426 a, 426 b may be applied by printing on the rear surface412, though they may be formed in or on the rear surface by othermethods, including laser marking.

FIG. 11( b) shows a transparent or translucent embossable layer 415applied to the front surface 411 of the transparent or translucentsubstrate 401. Optionally, the embossable layer is a radiation curableliquid, resin or ink, which may be applied by a printing process. Thelayer 415 is then embossed with an embossing shim 416 (FIG. 11( c)) toform a plurality of lenslets 422 of the lens array 420 in the layer 415in register with the image elements 426 a, 426 b on the rear surface 412of the substrate 410. The embossed layer 415 may be cured by radiation,e.g. by UV, X-rays, electron beams or heat (IR), either simultaneouslyduring the embossing process or subsequently, to fix the embossedstructure of the lenslets 422 in the lens array 420.

Referring to FIG. 12 there is shown an alternative method for producingan article 500 similar to that of FIG. 11( d) in that it is formed froma substrate 510 of transparent or translucent material having a lensarray 520 formed in an embossable layer 515 applied to a front surface511 of the substrate and image elements 526 a, 526 b formed in or on therear surface 512 of the substrate.

In the method shown in FIG. 12, the embossable layer 515 is firstapplied to the front surface 511 on one side of the substrate 510 (FIG.12( a)) and then embossed by embossing shim 516 (FIG. 12( b)) before theimage elements 526 a, 526 b are applied to the rear surface 512 on theopposite side of the substrate 510. Again, the embossable layer 515 maybe formed from a radiation curable liquid, resin or ink which may beapplied by a printing process, and may be cured by radiation eithersubstantially simultaneously during the embossing process or thereafter.The image elements 526 a, 526 b may be formed by printing or lasermarking on the rear surface 512 of the substrate 510 to form theresulting article 500 of FIG. 12( c).

In the resulting articles 400, 500 of FIG. 11( d) and FIG. 12( c), itwill be appreciated that the lens arrays 420, 520 have a gauge thicknesst=p+q, where p is the thickness of the transparent or translucentsubstrate 410, 510 and q is the thickness of the transparent ortranslucent layer 415, 515 measured from the front surface 411, 511 ofthe substrate 410, 510 to the apex of each lenslet 422, 522 afterembossing.

In many cases, the thicknesses p and q of the substrate 410 and layer415 will be predetermined, the average dot size H=2h will be determinedby the printing method or other process used to form the image elements,and one or more lens parameters, e.g. lens width W=2w, radius ofcurvature R, sag s, refractive index n or conic constant P may beoptimised in relation to t (=p+q) to create an embossing shim forforming the lens array 420, 520 in accordance with the process of FIG.14 described later.

Referring to FIGS. 13( a) to 13(d) there is shown a method for producingan article 600 which has a lens array 620 formed in a transparent ortranslucent layer 615 applied over image elements 626 a, 626 b on afront surface 611 of one side of a substrate 610. The substrate 610 inFIG. 13 may be transparent, translucent or opaque since the lens array620 and the image elements 626 a, 626 b are formed on the same side ofthe substrate 610. In the method shown in FIG. 13, the image elements626 a, 626 b are first applied to the front surface 611 on thesubstrate, optionally by printing (FIG. 13( a)), before the transparentor translucent layer 615 is applied (FIG. 13( b)) and embossed byembossing shim 616. Once again, the embossable layer 615 may be formedfrom a radiation curable liquid, resin or ink, which may be applied by aprinting process, and is cured by radiation either simultaneously orsubsequently to fix the lens structure of the lenslets 622 of the lensarray 620.

The resulting article 600 of FIG. 13( d) differs from that of FIGS. 11(d) and 12(c) in that the thickness p of the substrate 610 has no effecton the gauge thickness t of the lens array 620 which is substantiallyequal to the thickness q of the transparent or translucent layer 615(allowing for the thickness of the image elements 626 a, 626 b). As thegauge thickness t of the lens array 620 of article 600 is likely to beless than the gauge thickness t of the lens arrays 420, 520 of articles400, 500 of FIGS. 11( a) and 12(a), the method of one disclosedembodiment may be used to compensate for the reduced gauge thickness byreducing the lens width W, or radius of curvature R, or adjusting otherparameters of the lenslets 622 of the lens array 620, by appropriatevariation of the shape of the embossing shim 616.

Referring now to FIG. 14, there is shown a flow chart for a process forcreating an embossing shim for use with certain disclosed embodiments.Firstly, a calibration template is printed (step 700), and the dot sizemeasured (step 710) as described above. Then an initial set of lensparameters is chosen (step 720) and the parameters varied in amultivariate optimisation process (steps 730, 740). Once a solution isfound, an embossing shim can be created (step 750) for use in thefabrication process.

In FIGS. 15( a) and 15(b), flow charts of two alternative methods forforming an optical effect article are shown. In both cases a substrateis provided (step 800). The process shown in FIG. 15( a) is suitable forforming the articles 400 and 600 of FIGS. 11 and 13. In the embodimentof FIG. 15( a), two or more interleaved images are applied to a front orrear surface of the substrate (step 810), optionally by printing. Aradiation curable ink may then be applied to the front surface of thesubstrate (step 820), for example by a printing process, and the ink isthen embossed with an embossing shim obtained from step 750 of FIG. 14.The ink is then cured to form the lenslets of the optical effect articlein the embossed surface. The curing step may occur substantiallysimultaneously with the embossing step (step 830). In FIG. 15( b), theradiation curable ink is instead applied to one side of the substratefirst (step 840). The ink is then embossed with an embossing shimobtained from step 750 of FIG. 14 and cured to form the lenslets (step850). The image elements are then applied to the side of the substrateopposite the lenslets, in register with the lenslets, in order to formthe optical effect article.

EXAMPLE

Referring to FIG. 16, there is shown an example of a printed interleavedimage 900 which is used to produce a binary “flipping image” effect whenpaired with a suitable lens array. In the example shown, the imageelements are black stripes 901 interleaved with white stripes 902. Whenviewed through a lenticular lens array having lenticules 920, a devicehaving a combination of lenticular array and image elements 901, 902produces a switch from the image 910 shown at upper left in FIG. 16 toimage 920, in which black and white areas are reversed, as the device istilted with respect to the viewer about an axis parallel to thedirection of the stripes.

The black and white stripes 901, 902 were applied to a substrate bygravure printing. Upon measurement using the reticle of a microscope,the black stripes were found to have an average width of 32 microns,while the white stripes had an average width of 31.5 microns. Theaverage value of 32 microns for the black stripes was taken to be thescale parameter representative of the size of the image elements. Thewidth W of the lenticules 930 (shown overlaid in outline on the printedimage elements 901, 902) was fixed at 63.5 microns, and the gaugethickness t optimised using the expression in Equation (2). Thisresulted in an optimum gauge thickness t of 90 microns at a sag height sof 10 microns and a radius of curvature R of 55.4 microns, compared to agauge thickness of approximately 162 microns if the image elements werelocated at the nominal focal length of the lenticules.

In order to check that the focal point sizes of lenticules having theabove design were sufficiently close to the image element size toproduce the desired flipping image effect, the above parameters wereinput to a ray tracing simulation in optical system design softwareproduced by Zemax Development Corporation and sold under the trade markZEMAX. The relative illumination plots shown in FIGS. 17( a) and 17(b)can be used to determine the focal point size, this being the distancebetween the pairs of points (960 a, 960 b) and (961 a, 961 b)respectively where the relative illumination drops to zero. It can beseen that the on-axis focal spot size 951 is approximately 30 micronswhile the off-axis focal spot size 952 is approximately 23 microns. An“average” image element viewed on axis will thus be within 6%-7% of thefocal spot size.

It will be appreciated that various modifications may be made to thedisclosed embodiments described above without departing from the spiritor scope of the invention. For example, it is possible that lensletstructures of a lens array could be directly embossed into a surface ofa transparent or translucent substrate, instead of into a transparent ortranslucent embossable layer applied to a substrate. Also, whilstprinting one disclosed process for forming image elements, the imageelements may be formed by laser marking. In this case, it is possiblefor a laser to be directed through a transparent or translucentsubstrate or layer from a laser source on one side of the substrate orlayer to mark a laser sensitive surface on the opposite side of thesubstrate or layer to form the image elements after the lens array hasbeen formed.

1. A lens array for imaging a plurality of image elements in an objectplane, the lens array comprising: a plurality of lenslets formed in oron one side of a transparent or translucent material with the imageelements disposed on the opposite side, wherein the lens array has agauge thickness corresponding to a distance from an apex of eachlenslet, of the plurality of lenslets, to the object plane, wherein eachlenslet has a set of lens parameters, wherein at least one of the gaugethickness at least one lens parameter of the set of lens parameters isoptimised such that either: each lenslet of the plurality of lensletshas an effective focal width in the object plane which is substantiallyequal to the size of the image elements in the object plane, or eachlenslet of the plurality of lenslets has an effective focal width in theobject plane which varies from the size of the image elements by apredetermined amount, wherein the plurality of image elements include atleast two interlaced arrays of image elements.
 2. The lens array ofclaim 1, wherein the set of lens parameters for the lenslets include twoor more of the following: lens width, refractive index, sag height,radius of curvature, conic parameter and Abbe number.
 3. The lens arrayof claim 1, wherein the predetermined amount by which the effectivefocal width varies from the size of the image elements is not more than20% of the size of the image elements.
 4. The lens array of claim 1,wherein the set of lens parameters is the same for each of the lenslets.5. The lens array of claim 1, wherein the lenslets in one area or areasof the lens array have different lens parameters from the lenslets inanother area or other areas of the lens array.
 6. The lens array ofclaim 1, wherein the effective focal width, when averaged over two ormore directions within the lobe angle of the lenslet, is substantiallyequal to the size of the image elements, or varies from the size of theimage elements by a predetermined amount.
 7. A method of manufacturing alens array for imaging a plurality of image elements in an object plane,the lens array including a plurality of lenslets, the lens array havinga gauge thickness corresponding to the distance from the apex of eachlenslet to the object plane, the method comprising: determining a scaleparameter which is representative of the size of the image elements inat least part of the object plane, using the scale parameter to optimiseat least one of the gauge thickness and at least one parameter of a setof lens parameters for each lenslet, and forming the lens array with thegauge thickness and the lens parameters in or on one side of atransparent or translucent material with the image elements beingdisposed on an opposite side of the transparent or translucent material,whereby; each lenslet of the plurality of lenslets has an effectivefocal width in the object plane which is substantially equal to the sizeof the image elements, or each lenslet of the plurality of lenslets hasan effective focal width in the object plane which varies from the sizeof the image elements by a predetermined amount, wherein the pluralityof image elements include at least two interlaced arrays of imageelements.
 8. The method of claim 7, wherein the gauge thickness of thelens array is optimised with respect to the size of the image elementsand the set of lens parameters.
 9. The method of claim 7, wherein thelens parameters are optimised with respect to the size of the imageelements and the gauge thickness.
 10. The method of claim 7, wherein thepredetermined amount by which the effective focal width varies is lessthan an estimated variability in the size of the image elements, and theestimated variability is the standard deviation, median absolutedeviation or interquartile range of the size of the image elements. 11.The method of claim 7, wherein the image elements are applied to asurface on the opposite side of the transparent or translucent materialby printing, or by laser marking.
 12. The method of claim 7, wherein thelenslets are formed by an embossing process in a transparent ortranslucent radiation-curable material applied to a substrate formedfrom a transparent or translucent material, and the combined thicknessof the substrate and radiation-curable material corresponds to the gaugethickness of the lens array.
 13. A method of designing a lens array forimaging a plurality of image elements in an object plane the lens arrayincluding a plurality of lenslets and having a gauge thicknesscorresponding to a distance from an apex of each lenslet to the objectplane, the method comprising: estimating a scale parameter which isrepresentative of a size of each of the image elements of the imageelements in the object plane, selecting a set of lens parameters foreach lenslet of the plurality of lenslets, and designing the lens arrayusing the scale parameter to optimise at least one of the gaugethickness and at least one lens parameter of the set of lens parametersfor each lenslet of the plurality of lenslets, wherein: each lenslet ofthe plurality of lenslets has an effective focal width in the objectplane which is substantially equal to the size of the image elements, oreach lenslet of the plurality of lenslets has an effective focal widthin the object plane which varies from the size of the image elements bya predetermined amount, wherein the plurality of image elements includeat least two interlaced arrays of image elements.
 14. The method ofclaim 13, wherein the set of lens parameters includes two or more of thefollowing: lens width, refractive index, sag height, lens width, radiusof curvature, conic parameter and Abbe number.
 15. The method of claim13, further including measuring the sizes of the image elements in atleast part of the object plane, wherein the scale parameter is estimatedfrom the measured sizes of the image elements.
 16. A method ofmanufacturing an optically variable device, the method comprising:providing a substrate; applying image elements to the substrate, theimage elements being located in an object plane; determining a scaleparameter which is representative of the size of the image element; andforming a plurality of lenslets in a transparent or translucent materialon the substrate, wherein each lenslet of the plurality of lenslets hasa set of lens parameters determined so that: each of the lenslets of theplurality of lenslets has an effective focal width in the object planewhich is substantially equal to the size of the image elements, or eachof the lenslets of the plurality of lenslets has an effective focalwidth in the object plane which varies from the size of the imageelements by a predetermined amount, wherein the plurality of imageelements include at least two interlaced arrays of image elements. 17.The method of claim 16, wherein the scale parameter is determined bymeasuring the sizes of the image elements.
 18. An optically variabledevice, comprising: a substrate, and a plurality of lenslets formed inor on the substrate; and a plurality of image elements located in anobject plane in or on the substrate, wherein each lenslet of theplurality of lenslets has a set of lens parameters determined so that:each of the lenslets of the plurality of lenslets has an effective focalwidth in the object plane which is substantially equal to the size ofthe image elements; or each of the lenslets of the plurality of lensletshas an effective focal width in the object plane which varies from thesize of the image elements by a predetermined amount, wherein theplurality of image elements include at least two interlaced arrays ofimage elements.
 19. The device of claim 18, wherein the lenslets arepart of a lens array having a gauge thickness which is less than thefocal length of each of the lenslets.
 20. A security document includingan optically variable device of claim 18, wherein the optically variabledevice is formed in a window or half-window area of the securitydocument.
 21. A method of manufacturing a display device, comprising thesteps of: providing m images of an object, wherein m is at least equalto 2, dividing each image into sets of adjacent arrays of pictureelements, spaced at a mutual distance, applying the images in aninterlaced manner on an image layer in sets of interlaced arrays below alens structure comprising line-shaped lens elements over the image layerwith one line shaped lens element overlying a corresponding set ofadjacent arrays, wherein upon applying the arrays onto the image layer,and/or upon providing the lens elements, each array of picture elementsis provided onto the image layer in an out-of-focus manner to form ablurred array, or each array is imaged by the lens elements to form ablurred array, wherein a mutual distance of the edges of adjacentblurred arrays is smaller than the mutual distance by which the adjacentarrays of picture elements are spaced.
 22. The method of claim 21,wherein the array of picture elements is provided onto the image layerin an out-of-focus manner by means of printing.
 23. The method of claim21, wherein the array of picture elements is provided onto the imagelayer by laser marking.
 24. The method of claim 21, wherein the edges ofadjacent blurred arrays are substantially touching.
 25. The method ofclaim 21, wherein the increase in width caused by applying the arrays inan out-of-focus manner comprises between 0% and
 20. 26. A display devicecomprising an array of lens elements overlying an image layer with setsof interlaced arrays produced by the method of claim
 21. 27. A displaydevice comprising an image layer, a substrate having at a distance tabove the image layer a number of line-shaped lens elements which areadapted to focus a light beam for applying arrays of picture elementsonto the image layer in a focal plane situated at distance f from thelens-elements in which f is larger or smaller than t.
 28. The displaydevice of claim 27, where t is between 90 μm and 100 μm, f beingdifferent from H by up to 20%.